Show commands:
Magma
magma: G := TransitiveGroup(21, 138);
Group action invariants
Degree $n$: | $21$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $138$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_3^6.(C_2\times S_7)$ | ||
Parity: | $-1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,16,2,18,3,17)(4,20,5,21,6,19)(7,9)(10,12)(13,15), (1,17,5,7,13,2,16,6,9,14)(3,18,4,8,15)(10,21,12,20,11,19) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $5040$: $S_7$ $10080$: $S_7\times C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: None
Degree 7: $S_7$
Low degree siblings
42T2539, 42T2540, 42T2541Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
There are 118 conjugacy classes of elements. Data not shown.
magma: ConjugacyClasses(G);
Group invariants
Order: | $7348320=2^{5} \cdot 3^{8} \cdot 5 \cdot 7$ | magma: Order(G);
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Cyclic: | no | magma: IsCyclic(G);
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Abelian: | no | magma: IsAbelian(G);
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Solvable: | no | magma: IsSolvable(G);
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Nilpotency class: | not nilpotent | ||
Label: | 7348320.a | magma: IdentifyGroup(G);
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Character table: not available. |
magma: CharacterTable(G);